To differentiate him from the "cook", which is what we call those who carry out the actual act of cooking.

Well, don’t go around calling me a compiler!

If that's what you do, then the name is perfectly apt. Why shy away from what you are?

The difference is that the head chef can cook very well and could do a better job of the dish than the trainee.

"head chef" is a managerial position but yes often they can and do cook.